L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s − 2·11-s − i·12-s + 4i·13-s + 16-s − 6i·17-s + i·18-s + 8·19-s + 2i·22-s − i·23-s − 24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.603·11-s − 0.288i·12-s + 1.10i·13-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s + 1.83·19-s + 0.426i·22-s − 0.208i·23-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.482737436\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.482737436\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.108792932344571634927519014254, −7.906007519820990042421048372561, −7.34789011723112045438592156963, −6.33198267606459058781549834071, −5.20999647729956396555254722645, −4.90337421142677275128985205570, −3.87611049359390698065887618396, −3.07629251146014194041135800566, −2.27947879961208929765099071462, −0.945099906505614831662834854500,
0.56485119263180306817068232408, 1.80728213568130872542643341006, 3.08154558190132171322615153754, 3.79431825527151714091635283865, 5.10195536559024912931784383956, 5.58876043468952255389389342209, 6.23059551650532439949833130639, 7.29623657180821848906325632604, 7.64364110372990049015430630430, 8.323516246670792470470583458685